Find the Asymptotes (x^2-64)/(x^4-81)
Brief Explanation:
The problem asks to determine the asymptotes of the rational function provided, which is (x^2-64)/(x^4-81). Asymptotes are lines that the graph of the function approaches but does not cross as the values of x become large in the positive or negative direction. There are generally three types of asymptotes to consider for rational functions:
Vertical asymptotes occur where the denominator of the rational function is equal to zero. This is where the function is undefined, and the graph typically goes to positive or negative infinity.
Horizontal asymptotes are found by examining the end behavior of the function as x approaches infinity or negative infinity. They occur when the degrees of the polynomials in the numerator and the denominator are the same or when the degree of the polynomial in the numerator is less than that in the denominator.
Oblique or slant asymptotes may occur when the degree of the numerator is exactly one more than the degree of the denominator. In such cases, you may need to perform polynomial long division to find the slant asymptote.
The question is asking for a determination of all types of asymptotes for the given rational function, which would involve analyzing its numerator and denominator and their respective degrees and zeros.
$\frac{x^{2} - 64}{x^{4} - 81}$
Answer
Solution:
Step 1:
Determine the values of $x$ for which the function $\frac{x^2 - 64}{x^4 - 81}$ does not exist. These are $x = -3$ and $x = 3$.
Step 2:
Analyze the behavior of the function as $x$ approaches $-3$. The function tends to negative infinity from the left and positive infinity from the right, indicating a vertical asymptote at $x = -3$.
Step 3:
Examine the behavior of the function as $x$ approaches $3$. The function tends to positive infinity from the left and negative infinity from the right, confirming a vertical asymptote at $x = 3$.
Step 4:
Compile a list of all vertical asymptotes found: $x = -3$ and $x = 3$.
Step 5:
Review the rules for determining horizontal asymptotes for the function $R(x) = \frac{ax^n}{bx^m}$, where $n$ is the degree of the numerator and $m$ is the degree of the denominator. The horizontal asymptote depends on the relationship between $n$ and $m$.
Step 6:
Identify the degrees $n$ and $m$ for the given function. Here, $n = 2$ and $m = 4$.
Step 7:
Since the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the horizontal asymptote is the x-axis, given by $y = 0$.
Step 8:
Conclude that there are no oblique asymptotes, as the degree of the numerator is not greater than the degree of the denominator.
Step 9:
Summarize all asymptotes of the function:
- Vertical Asymptotes: $x = -3$, $x = 3$
- Horizontal Asymptote: $y = 0$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, one must understand the different types of asymptotes and the conditions under which they occur:
Vertical Asymptotes: These occur at values of $x$ where the denominator of a rational function is zero, and the numerator is not zero at those points. The function will approach infinity or negative infinity as $x$ approaches these values from either side.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and the denominator ($n$ and $m$ respectively). If $n < m$, the horizontal asymptote is $y = 0$. If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator. If $n > m$, there is no horizontal asymptote.
Oblique Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. In such cases, one can perform polynomial long division to find the equation of the oblique asymptote.
In the given problem, the vertical asymptotes are found by setting the denominator equal to zero and solving for $x$. The horizontal asymptote is determined by the degrees of the numerator and denominator. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$. There are no oblique asymptotes because the degree of the numerator is not one more than the degree of the denominator.
